We provide formal matched asymptotic expansions for ancient convex
solutions to MCF. The formal analysis leading to the solutions is
analogous to that for the generic MCF neck pinch in
[1].
For any $p, q$ with $p+q=n$, $p\geq1$, $q\geq2$ we find a formal ancient
solution which is a small perturbation of an ellipsoid. For
$t\to-\infty$ the solution becomes increasingly astigmatic: $q$ of its
major axes have length $\approx\sqrt{2(q-1)(-t)}$, while the other $p$ axes
have length $\approx \sqrt{-2t\log(-t)}$.
We conjecture that an analysis similar to that in [2] will
lead to a rigorous construction of ancient solutions to MCF with the
asymptotics described in this paper.
Citation: Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovalsin mean curvature flow[J]. Networks and Heterogeneous Media, 2013, 8(1): 1-8. doi: 10.3934/nhm.2013.8.1
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Abstract
We provide formal matched asymptotic expansions for ancient convex
solutions to MCF. The formal analysis leading to the solutions is
analogous to that for the generic MCF neck pinch in
[1].
For any $p, q$ with $p+q=n$, $p\geq1$, $q\geq2$ we find a formal ancient
solution which is a small perturbation of an ellipsoid. For
$t\to-\infty$ the solution becomes increasingly astigmatic: $q$ of its
major axes have length $\approx\sqrt{2(q-1)(-t)}$, while the other $p$ axes
have length $\approx \sqrt{-2t\log(-t)}$.
We conjecture that an analysis similar to that in [2] will
lead to a rigorous construction of ancient solutions to MCF with the
asymptotics described in this paper.
References
|
[1]
|
S. B. Angenent and J. J. L. Velázquez, Degenerate neckpinches in mean curvature flow, J. Reine Angew. Math., 482 (1997), 15-66.
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[2]
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Sigurd Angenent, Cristina Caputo and Dan Knopf, Minimally invasive surgery for Ricci flow singularities, J. Reine Angew. Math., 672 (2012), 39–87(to appear).
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[3]
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Sigurd Angenent, Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), Progr. Nonlinear Differential Equations Appl., 7 Birkhäuser Boston, Boston, MA, (1992), 21-38.
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[4]
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Panagiota Daskalopoulos, Richard Hamilton and Natasa Sesum, Classification of compact ancient solutions to the curve shortening flow, J. Differential Geom., 84 (2010), 455-464.
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[5]
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M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.
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